#200157

order	512
nilpotency class	3
isoclinism	isoclinic class of #200152
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60954	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

977 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,12)(4,17,20,9)(5,10)(6,21)(7,11)(8,15)(13,26,30,32)(14,23)(16,27)(18,25,19,24)(22,29,31,28)(33,34)(35,39,44,53)(36,46,47,58)(37,42)(38,41)(40,48,55,50)(43,59)(45,54,49,57)(51,56)(52,62)(60,63)(61,64);(1,3,6,15,7,16,5,14)(2,8,10,12,11,23,21,27)(4,13,18,28,20,30,19,29)(9,22,25,26,17,31,24,32)(33,35)(34,39,41,53)(36,43)(37,49)(38,44)(40,52,55,64)(42,57,56,54)(45,51)(46,63,58,59)(47,60)(48,62)(50,61);(1,4)(2,9)(3,13)(5,19)(6,18)(7,20)(8,22)(10,25)(11,17)(12,26)(14,29)(15,28)(16,30)(21,24)(23,31)(27,32)(33,36)(34,40)(35,43)(37,50)(38,47)(39,52)(41,55)(42,58)(44,60)(45,62)(46,56)(48,51)(49,61)(53,64)(54,63)(57,59);(1,5,7,6)(2,10,11,21)(3,14,16,15)(4,18,20,19)(8,12,23,27)(9,24,17,25)(13,28,30,29)(22,32,31,26)(34,41)(36,47)(39,53)(42,56)(43,60)(48,50)(54,57)(61,62)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107353);